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Journal of Modern Transportation
Volume 19, Number 1, March 2011, Page 58-67
Journal homepage: jmt.swjtu.edu.cn
DOI: 10.1007/BF03325741
In-depth analysis of traffic congestion using computational
fluid dynamics (CFD) modeling method
1* 1 2
Dazhi SUN , Jinpeng LV , S. Travis WALLER
1. Department of Civil & Architectural Engineering, Texas A&M University-Kingsville, Texas 78363, USA
2. Department of Civil Engineering, The University of Texas at Austin, Austin, Texas 78712, USA
Abstract: This paper introduces computational fluid dynamics (CFD), a numerical approach widely and successfully
used in aerospace engineering, to deal with surface traffic flow related problems. After a brief introduction of the
physical and mathematical foundations of CFD, this paper develops CFD implementation methodology for modeling
traffic problems such as queue/platoon distribution, shockwave propagation, and prediction of system performance.
Some theoretical and practical applications are discussed in this paper to illustrate the implementation methodology. It
is found that CFD approach can facilitate a superior insight into the formation and propagation of congestion, thereby
supporting more effective methods to alleviate congestion. In addition, CFD approach is found capable of assessing
freeway system performance using less ITS detectors, and enhancing the coverage and reliability of a traffic detection
system.
Key words: CFD; Euler’s equation; shockwaves; queue/platoon; system performance monitoring
© 2011 JMT. All rights reserved.
1. Introduction Computational fluid dynamics (CFD) began through
investigation by Harlow in 1956 focusing on the move-
t has been over a half century since engineers and ment of fluid materials under high compression [8]. In
I experts incorporated the theory of fluid dynamics in 1981, the first general purpose CFD package,
to transportation study. It began in the 1950’s when PHOENICS, was developed and released by Concentra-
Lighthill and Whitham [1] introduced a one-dimensional tion Heat and Momentum Limited (CHAM) [9]. How-
method, which allowed for the study of transportation ever, as an important numerical method, CFD has not
problems using fluid dynamic method. Later, Richards yet been implemented to solve the traffic flow problems.
[2] developed a simple traffic flow under the precondi- So far, only an implementation in determining piping in
tion that the movement of a group of discrete vehicles the transportation field has been reported [10]. This pa-
could be treated as a continuous flow and and the equa- per will focus on examining the application of CFD
tion of the conservation of matter was given as method to traffic flow analysis.
For Euler’s equation implementations in transporta-
ddv tion engineering, there are traditionally two viewpoints.
0, (1) The first is referred to as the Lagrangian description.
tx
which is also called the one-dimensional Euler’s equa- This method concentrates on individual particles in a
tion in fluid mechanics. fluid flow, or individual vehicles in traffic study. La-
Since these early pioneering works, the study and use grangian description has been applied for studying cer-
of the one-dimensional Euler’s equation in traffic flow tain traffic flow problems, such as car-following studies.
theory has continued to be a topic of interest [3-6]. Re- When utilizing the Eulerian viewpoint, instead of indi-
cently, for example, Laval and Daganzo proposed an ef- vidual vehicles in a flow, traffic is viewed as a simple
fective implementation of the one-dimensional Euler’s continuously distributed flow, with consistent gaps be-
equation for lane-changing study [7]. tween the cars constituting various levels of density,
with more emphasis on given road segments. The meth-
odology presented in this paper is based on the Eulerian
description, and thus emphasis was placed on the flow
Received Dec. 23, 2010; revision accepted Jan. 14, 2011
*Corresponding author. Tel.: +1-361-593-2270 as a whole or a system and not on the individual
Email: kfds000@tamuk.edu (D.Z. SUN) vehicles.
doi: 10.3969/j.issn.2095-087X.2011.01.009
Journal of Modern Transportation 2011 19(1): 58-67 59
2. Methodology k kv
div( ) 0. (4)
t
DOI: 10.1007/BF03325733
This section will describe the fundamental methodo- In traffic flow, m is the number of vehicles on a road.
logical steps related to the CFD approach. First the basic The generalized equation (4) works not only in one-
principles of CFD will be discussed followed by traffic- dimensional linear roadway segment in most situations
specific implementation issues. (Fig. 1(b)), but also in two-dimensional problems
(Fig. 1(c)).
2.1. Fundamentals of CFD The generalized equation (Eq. (4)) is simplified to the
1D condition:
First, the one-dimensional Euler’s equation will be
deduced and explored. Let C be a control volume (C kkv
does not change with time, Fig.1 (a)). Due to the con- 0. (5)
servation of mass, the rate of change of mass in C is tx
To implement the CFD approach, one necessary as-
ddkk sumption is that there exists some empirical relationship
mC(,t)(x,t)dV (x,t)dV,
C (2)
ddttt t between speed and density, that is, the relationship be-
where m is mass; k is density. tween the flow q and the density (concentration) k. This
relationship between q and k might vary with location x
Unit normal n but not with time t, i.e.,
*
qx(,t) q(k(,xt)). (6)
V *
For some given function q , the conservation equa-
tion develops into:
dA *
kx(,t) q(kx(,t)) (7)
0.
tx
Portion of the This equation now only has two independent factors,
boundary of C location x and time t.
When discussing the problem of a shock wave, New-
ell [5] emphasized the meaning of the relationship be-
*
(a) tween the two independent factors. If the location x is
*
given, the slope of q(x ,t) can be found. And when the
* *
time t is given, the slope of k(x,t ) can be found. The
discontinuity of the slope represents the shock wave. So
it is not necessary to track the actual path of the shock
(b) wave to determine the time at which a shock passes a
given location, or the location which a shock arrives at a
given time.
Typically, it is difficult to obtain a perfect mathe-
matical solution to partial differential equations. There-
fore, numerical solution methods have been widely used;
however, the initial and boundary conditions need to be
(c) specified beforehand. The foundation of numerical
Fig. 1 The Eulerian description methods is the Taylor formula:
The mass crossing the boundary C per unit time 22
ktk
nn1
kk
equals the surface integral of over . The prin- ii 2
kvn C tt
2
ciple of conservation of mass can be more precisely 33 44
tktk
5
stated as: ot(), (8)
34
62tt4
d 22
ktk
kx(,t)dV kvn d.A (3) nn1
CC kk
dt ii 2
tt2
33 44
Because this is true for all C, it is therefore equivalent tktk 5
ot(), (9)
34
to 62tt4
60 Dazhi SUN et al. / In-depth analysis of traffic congestion using computational fluid dynamics (CFD) …
2 2 As described in (6), if there exists some relationship
qq
nn x *
qqx
ii1
x 2 x2 q between q and k, computational algorithms can be
DOI: 10.1007/BF03325733 employed to calculate the parameters for a given road.
34
34
xxqq
5 (10)
ox(),
To keep solutions stable, the following constraint condi-
34
624
xx
tion is required:
2 2
qq
nn x
qqx
1
ii x 2 t
2 x
k 1. (18)
3434 max x
qq
xx
5 (11)
ox(),
34
624
xx
n n 2.2. Implementation methodology of CFD in traffic flow
where o denotes the error; q and k are the traffic vol-
i i
ume and the density when x= and t= .
ix nt For numerical computation the basic concepts de-
In this step, although location x and time t are step scribed previously can be deployed via the following
functions, if x and t are small enough, and the re- simple steps. First, the initial condition must be given,
sults deduced are accurate enough for the transportation which is the initial density. The iterations can then be
problems, the location x and time t can be still treated as started. Finally, conditions are applied to terminate the
continuous. Taylor formula can be changed to a differ- iterative procedure. The process can generate various
ence format. For forward difference, outputs depending on the research requirements.
nn1 2
kk
ktk
ii
2.2.1. Study the path of shock waves
2 2
ttt
23 34
tktk 5 *
If we want to know at a given time t = where the
ot(). (12) nt
34
62tt4 shock wave is, we can adopt the stop condition, and
nn2
qq
qq
x
ii1 output the densities of any location x. The location of
i
xx2x2 the discontinuity of the density is the location of the
2334
qq
xx 5 shock wave.
ox().
(13)
34
624
xx
On the other hand, if we want to know at a given lo-
*
For backward difference, cation x =jx when the shock wave will arrive, we can
adopt the stop condition:
nn1 22
kk
ktk
ii
xx 澶. (19)
2 jj1
tt2 t
33 44 In practice, the programmer usually adopts x (a is a small
tktk 5 j-a
ot(), (14)
34 integer). Consider the shakes in the location of the discontinu-
62tt4
nn 22 ous point, and output the number of iterations n. is the
qq nt
qxq
ii1
2 time that the shock wave reaches the given location.
xx2x
33 44
xqxq 5 (15)
ox().
2.2.2. Estimate traffic volume
34
62xx4
Using the difference formats of the Taylor formula, The traffic volume is easily calculated by the given
Eq. (7) is changed into the following format: *
function q after acquiring the density in any location. We
The difference of k+the difference of q=0, (16) can use the volumes of the input and output of a region to
calculate the number of vehicles within the region. Con-
where the items in the differences with high order small tinuously monitoring this parameter can help identify
amount t or x will be ignored. whether there is a breakdown in this region, even if no
To maintain sufficient accuracy, forward difference is data about density or volume is collected within this re-
applied first, followed by backward difference (which is gion [11]. First, the initial condition is needed. The initial
at times referred to as MacCormack’s method): number of vehicles N can be calculated as:
0
t je
nn1 nn 0
kkqq
(), Nkx,
iiii1 0 i (20)
x ij
t
nn11 n1n1
kk qq
(), (17) 0 0
iix i1 i where kj is the density in the input of the region; k j+e is
the density in the output of the region; is the length
1 ex
nn11n
kk()k.
iii of the region on the road. There are two methods to cal-
2
Journal of Modern Transportation 2011 19(1): 58-67 61
culate the time and the number of vehicles N . One From the relationship between the time and the dis-
nt n
employs the density, which is similar to (20): tance, the T-S diagram can then be drawn.
DOI: 10.1007/BF03325733
je
n 2.2.4. A sample implementation
Nkx, (21)
ni
ij
n In the following examples, a simplified relationship
ek is the density at the time .
i nt between the velocity v and the density k is adopted:
The other, employs the in and out volumes of a given
region. During the time , the number of vehicles that vk2.5 100, (27)
nt
2
traveled into and out of the region are: qk2.5 100k. (28)
n n Problem description:
Nqt,
in j
0 On a road which is 10 miles long, there exist two one
n n mile long queues in the location x=2 mile and x=6 mile,
Nqt.
out je respectively (Fig. 2). Note, the locations with elevated
0 density represent queues. In the first part of the road, the
Therefore, at the time , the number of vehicles N is:
nt n lower density is 15 vehicles per mile (vpm), and the
NNNN higher density is 30 vpm. The volume is the maximum
n 0 in out
je nn at the location seven miles away from the starting point,
0 nn
kx qt qt. (22)
where the density is
ijje
ij 00 k
max vpm.
k
The computational workload depends on the length 2 20
of time and the length of the region. To reduce the com- 40
putational workload, method 1 is preferred for longer
time periods; otherwise method 2 is better for longer
road regions. 30
)
2.2.3. Study the time-space diagram m
p 20
(v
ity
The time-space diagram has been widely used for s
n
solving some traffic-related problems such as gap stud- e 10
D
ies. In some time t , a vehicle is in the location S where
0 0
the density is k . Then, the velocity v in this location
1 1 0
can be computed through the relationship between q and 0 2 4 6 8 10
k (since q=kv). Next, the distance traveled by the vehicle Location (mile)
during the first time segment t is calculated with the Fig. 2 The initial condition of the example
equation Problems:
(for the first iteration of time)ˈ
vt (23)
1 (1) Present the movement of traffic flow waves.
and the location in this time is (2) Detect the number of vehicles in a 2-mile long re-
gion from x=3 mile to x=5 mile.
S =S . (24)
1 0+vt (3) Plot the trajectories of two selected vehicles at the
1
In the next time segment t +t , the density k is de- location x=0 mile and x=2 mile.
m 2 Solution:
termined in the location S , and then the velocity v is
1 2 From (18),
computed in this location through the relationship be- t
tween q and k. Therefore, the distance traveled by the kmax x 1, kmax 40vpm.
vehicle during the second time segment t and the lo- Choose t=0.000 1, x=0.01 and then
cation can be calculated:
v2t (for the second iteration of time), k t 0.0001
max 40 0.4 1,
S =S v t. (25) x 0.01
2 1+ 2 which implies that the road is divided into 1 000 sec-
Ă tions, and the accuracy of time is 0.000 1 h.
By analogy, after m segments of time t , (1) For the termination condition of the iterative pro-
S =S +v t. (26) n+1 n
m m- cedure, this research adopts that when k -k ˘0.01,
1 m i i
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